Unvalued directed graph with 20 vertices, all interconnected. There is a geodesic distance of one from any vertex to any other vertex. Density=1.0. 

Contexts in source publication

Context 1

... of the application of social network analysis to interregional flows are not plentiful, but one can find work on trade flows among nations (Snyder and Kick 1979; Kick and Davis 2001) or among major world cities (Shin and Timberlake 2000; Smith and Timberlake 2001). As will be shown, network analysis provides tools that can be used to delineate the hierarchical relations among regions. The following section introduces some ideas from social network analysis. The next section then discusses three data sources for interregional flows of labor and goods in the U.S. The paper then presents a technique for singling out from the myriad of interregional flows those that represent a region’s position in the central place hierarchy. The technique is then applied to six data matrices, and the results compared to determine the degree of congruence among the six derived hierarchies. The paper concludes with discussion of potential uses of the method. Interregional exchange networks can be presented as graphs. Figure 1 presents a view of how the interregional exchanges posited by Walter Christaller’s (1933) Central Place Theory would appear in graph format. Each node or vertex of the graph represents a region; each line or link represents a flow of goods or people; the system of regions has three levels, and each level is characterized by a hub and spoke structure. The single node at the center of the graph is the highest order center. Figure 2 presents a graph in which all of the regions exchange directly with each other. There is no hierarchical structure in this graph: each region has a comparative advantage and produces for exchange with every other region. Of course, both central place models and export base models are true, both describe some part of the reality of how regions are interrelated. In practice, this suggests that empirical graphs of interregional flows will look like Figure 2, and one must have some method of singling out those lines which constitute the central place structure. When lines have a direction, the graph is called a or digraph. When lines have a value, as they would when representing the volume of trade from one node to another, the graph is called a valued graph . The number of lines one must traverse to move from one node to another is the path length . If there is a path from node i to node j , then node i is reachable from node j . The shortest path length between any two nodes is the geodesic distance connecting those two nodes. Figure 1 shows the largest geodesic distance for each of the nodes; thus, for example, from the node in the center of the graph (representing the highest order central place) one can reach any other node in the graph by traversing at most two lines. From the peripheral nodes, ...

Context 2

... of the application of social network analysis to interregional flows are not plentiful, but one can find work on trade flows among nations (Snyder and Kick 1979; Kick and Davis 2001) or among major world cities (Shin and Timberlake 2000; Smith and Timberlake 2001). As will be shown, network analysis provides tools that can be used to delineate the hierarchical relations among regions. The following section introduces some ideas from social network analysis. The next section then discusses three data sources for interregional flows of labor and goods in the U.S. The paper then presents a technique for singling out from the myriad of interregional flows those that represent a region’s position in the central place hierarchy. The technique is then applied to six data matrices, and the results compared to determine the degree of congruence among the six derived hierarchies. The paper concludes with discussion of potential uses of the method. Interregional exchange networks can be presented as graphs. Figure 1 presents a view of how the interregional exchanges posited by Walter Christaller’s (1933) Central Place Theory would appear in graph format. Each node or vertex of the graph represents a region; each line or link represents a flow of goods or people; the system of regions has three levels, and each level is characterized by a hub and spoke structure. The single node at the center of the graph is the highest order center. Figure 2 presents a graph in which all of the regions exchange directly with each other. There is no hierarchical structure in this graph: each region has a comparative advantage and produces for exchange with every other region. Of course, both central place models and export base models are true, both describe some part of the reality of how regions are interrelated. In practice, this suggests that empirical graphs of interregional flows will look like Figure 2, and one must have some method of singling out those lines which constitute the central place structure. When lines have a direction, the graph is called a or digraph. When lines have a value, as they would when representing the volume of trade from one node to another, the graph is called a valued graph . The number of lines one must traverse to move from one node to another is the path length . If there is a path from node i to node j , then node i is reachable from node j . The shortest path length between any two nodes is the geodesic distance connecting those two nodes. Figure 1 shows the largest geodesic distance for each of the nodes; thus, for example, from the node in the center of the graph (representing the highest order central place) one can reach any other node in the graph by traversing at most two lines. From the peripheral nodes, ...

Context 3

... from the 1990 census gives inter-county migration counts for persons by age, sex, race, educational attainment, nativity, and poverty status. Thus, working age migrants can be singled out, as well as migrants with high education levels. These data can be aggregated to identify migration patterns at the NTAR level. Six directed, valued adjacency matrices are created from these data. Each matrix depicts the flows among 89 NTAR regions, where the flow from the row NTAR to the column NTAR is given by each element. Two matrices are of commuting data: Journey to Work 1980, and Journey to Work 1990. Two matrices are of migration data: all 1990 migrants from ages 25 to 69, and those 1990 migrants from ages 25 to 59 who have educational attainment of at least a Bachelor’s degree. Two matrices are of trade data: 1993 value of shipments, and 1993 weight of all shipments. Do these matrices represent graphs that look more like Figure 1 or Figure 2? One way to answer this question is to calculate graph density . A graph’s density is the number of lines between nodes divided by the number of pairs of nodes; it gives the percent of possible lines actually present in the graph. The density of Figure 1 is about 5%, and of Figure 2 100%. The density of the six graphs are given in Table 1. For a graph like Figure 1—with a single upward link for each node— the density of a network with 89 nodes would be 88/(89*88) = 1.12%. All of the matrices in fact represent graphs much denser than that, with the migration matrices having a density very close to 100%. Thus, the raw data clearly resemble Figure 2, but as we noted above, we would expect empirical relationships to resemble Figure 2 in a world where regions both fill a role on a central place hierarchy and exchange as a specialized unit with every other region. The task is to tease out from the myriad of links those links that represent the central place hierarchy. Flows among regions can be depicted in an x matrix , where the flow from region to region j is given by each element f ij . Flows can be converted to percentages, to dampen the effect of differential region sizes, in an n x n matrix P , where the flow from region i to region j is given by each element p ij = f ij / ∑ j f ij . The element-wise geometric mean of P and its transpose creates a symmetric matrix A , where each element a ij =( p ij p ji ) 1⁄2 . Setting the diagonal equal to zero, one can use A to calculate a centrality vector c ...

Context 4

... from the 1990 census gives inter-county migration counts for persons by age, sex, race, educational attainment, nativity, and poverty status. Thus, working age migrants can be singled out, as well as migrants with high education levels. These data can be aggregated to identify migration patterns at the NTAR level. Six directed, valued adjacency matrices are created from these data. Each matrix depicts the flows among 89 NTAR regions, where the flow from the row NTAR to the column NTAR is given by each element. Two matrices are of commuting data: Journey to Work 1980, and Journey to Work 1990. Two matrices are of migration data: all 1990 migrants from ages 25 to 69, and those 1990 migrants from ages 25 to 59 who have educational attainment of at least a Bachelor’s degree. Two matrices are of trade data: 1993 value of shipments, and 1993 weight of all shipments. Do these matrices represent graphs that look more like Figure 1 or Figure 2? One way to answer this question is to calculate graph density . A graph’s density is the number of lines between nodes divided by the number of pairs of nodes; it gives the percent of possible lines actually present in the graph. The density of Figure 1 is about 5%, and of Figure 2 100%. The density of the six graphs are given in Table 1. For a graph like Figure 1—with a single upward link for each node— the density of a network with 89 nodes would be 88/(89*88) = 1.12%. All of the matrices in fact represent graphs much denser than that, with the migration matrices having a density very close to 100%. Thus, the raw data clearly resemble Figure 2, but as we noted above, we would expect empirical relationships to resemble Figure 2 in a world where regions both fill a role on a central place hierarchy and exchange as a specialized unit with every other region. The task is to tease out from the myriad of links those links that represent the central place hierarchy. Flows among regions can be depicted in an x matrix , where the flow from region to region j is given by each element f ij . Flows can be converted to percentages, to dampen the effect of differential region sizes, in an n x n matrix P , where the flow from region i to region j is given by each element p ij = f ij / ∑ j f ij . The element-wise geometric mean of P and its transpose creates a symmetric matrix A , where each element a ij =( p ij p ji ) 1⁄2 . Setting the diagonal equal to zero, one can use A to calculate a centrality vector c ...

Context 5

... from the 1990 census gives inter-county migration counts for persons by age, sex, race, educational attainment, nativity, and poverty status. Thus, working age migrants can be singled out, as well as migrants with high education levels. These data can be aggregated to identify migration patterns at the NTAR level. Six directed, valued adjacency matrices are created from these data. Each matrix depicts the flows among 89 NTAR regions, where the flow from the row NTAR to the column NTAR is given by each element. Two matrices are of commuting data: Journey to Work 1980, and Journey to Work 1990. Two matrices are of migration data: all 1990 migrants from ages 25 to 69, and those 1990 migrants from ages 25 to 59 who have educational attainment of at least a Bachelor’s degree. Two matrices are of trade data: 1993 value of shipments, and 1993 weight of all shipments. Do these matrices represent graphs that look more like Figure 1 or Figure 2? One way to answer this question is to calculate graph density . A graph’s density is the number of lines between nodes divided by the number of pairs of nodes; it gives the percent of possible lines actually present in the graph. The density of Figure 1 is about 5%, and of Figure 2 100%. The density of the six graphs are given in Table 1. For a graph like Figure 1—with a single upward link for each node— the density of a network with 89 nodes would be 88/(89*88) = 1.12%. All of the matrices in fact represent graphs much denser than that, with the migration matrices having a density very close to 100%. Thus, the raw data clearly resemble Figure 2, but as we noted above, we would expect empirical relationships to resemble Figure 2 in a world where regions both fill a role on a central place hierarchy and exchange as a specialized unit with every other region. The task is to tease out from the myriad of links those links that represent the central place hierarchy. Flows among regions can be depicted in an x matrix , where the flow from region to region j is given by each element f ij . Flows can be converted to percentages, to dampen the effect of differential region sizes, in an n x n matrix P , where the flow from region i to region j is given by each element p ij = f ij / ∑ j f ij . The element-wise geometric mean of P and its transpose creates a symmetric matrix A , where each element a ij =( p ij p ji ) 1⁄2 . Setting the diagonal equal to zero, one can use A to calculate a centrality vector c ...

Context 6

... from the 1990 census gives inter-county migration counts for persons by age, sex, race, educational attainment, nativity, and poverty status. Thus, working age migrants can be singled out, as well as migrants with high education levels. These data can be aggregated to identify migration patterns at the NTAR level. Six directed, valued adjacency matrices are created from these data. Each matrix depicts the flows among 89 NTAR regions, where the flow from the row NTAR to the column NTAR is given by each element. Two matrices are of commuting data: Journey to Work 1980, and Journey to Work 1990. Two matrices are of migration data: all 1990 migrants from ages 25 to 69, and those 1990 migrants from ages 25 to 59 who have educational attainment of at least a Bachelor’s degree. Two matrices are of trade data: 1993 value of shipments, and 1993 weight of all shipments. Do these matrices represent graphs that look more like Figure 1 or Figure 2? One way to answer this question is to calculate graph density . A graph’s density is the number of lines between nodes divided by the number of pairs of nodes; it gives the percent of possible lines actually present in the graph. The density of Figure 1 is about 5%, and of Figure 2 100%. The density of the six graphs are given in Table 1. For a graph like Figure 1—with a single upward link for each node— the density of a network with 89 nodes would be 88/(89*88) = 1.12%. All of the matrices in fact represent graphs much denser than that, with the migration matrices having a density very close to 100%. Thus, the raw data clearly resemble Figure 2, but as we noted above, we would expect empirical relationships to resemble Figure 2 in a world where regions both fill a role on a central place hierarchy and exchange as a specialized unit with every other region. The task is to tease out from the myriad of links those links that represent the central place hierarchy. Flows among regions can be depicted in an x matrix , where the flow from region to region j is given by each element f ij . Flows can be converted to percentages, to dampen the effect of differential region sizes, in an n x n matrix P , where the flow from region i to region j is given by each element p ij = f ij / ∑ j f ij . The element-wise geometric mean of P and its transpose creates a symmetric matrix A , where each element a ij =( p ij p ji ) 1⁄2 . Setting the diagonal equal to zero, one can use A to calculate a centrality vector c ...